A076566 -id:A076566

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A000082

a(n) = n^2*Product_{p|n} (1 + 1/p).

+10
12

1, 6, 12, 24, 30, 72, 56, 96, 108, 180, 132, 288, 182, 336, 360, 384, 306, 648, 380, 720, 672, 792, 552, 1152, 750, 1092, 972, 1344, 870, 2160, 992, 1536, 1584, 1836, 1680, 2592, 1406, 2280, 2184, 2880, 1722, 4032, 1892, 3168, 3240, 3312, 2256

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

For n > 1: A006530(a(n)) = A076566(n-1). - Reinhard Zumkeller, Oct 03 2012

A strong divisibility sequence, that is, gcd(a(n), a(m)) = a(gcd(n, m)) for all positive integers n and m. - Michael Somos, Jan 01 2017

REFERENCES

B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 79.

LINKS

T. D. Noe, Table of n, a(n) for n = 1..1000

Index to divisibility sequences

FORMULA

a(n) = n * A001615(n).

Dirichlet g.f.: zeta(s-1)*zeta(s-2)/zeta(2*s-2).

Dirichlet convolution: Sum_{d|n} mu(n/d)*sigma(d^2). - Vladeta Jovovic, Nov 16 2001

Multiplicative with a(p^e) = p^(2*e-1)*(p+1). - David W. Wilson, Aug 01 2001

a(n) = A181797(n)*A003557(n). - R. J. Mathar, Mar 30 2011

a(n) = A001615(n^2). - Enrique Pérez Herrero, Mar 06 2012

Sum_{k=1..n} a(k) ~ 5*n^3 / Pi^2. - Vaclav Kotesovec, Jan 11 2019

Sum_{n>=1} 1/a(n) = A335762. - Amiram Eldar, Jun 23 2020

MAPLE

proc(n) local b, d: b := n^2: for d from 1 to n do if irem(n, d) = 0 and isprime(d) then b := b*(1+d^(-1)): fi: od: RETURN(b): end:

MATHEMATICA

Table[ Fold[ If[ Mod[ n, #2 ]==0 && PrimeQ[ #2 ], #1*(1+1/#2), #1 ]&, n^2, Range[ n ] ], {n, 1, 45} ]

Table[ n^2 Times@@(1+1/Select[ Range[ 1, n ], (Mod[ n, #1 ]==0&&PrimeQ[ #1 ])& ]), {n, 1, 45} ] (* Olivier Gérard, Aug 15 1997 *)

f[p_, e_] := (p+1)*p^(2*e - 1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Jun 23 2020 *)

PROG

(PARI) a(n)=if(n<1, 0, direuler(p=2, n, (1+p*X)/(1-p^2*X))[n])

(Haskell)

a000082 n = product $ zipWith (\p e -> p ^ (2*e - 1) * (p + 1))

(a027748_row n) (a124010_row n)

-- Reinhard Zumkeller, Oct 03 2012

CROSSREFS

Cf. A001615, A027748, A033196, A124010, A335762.

KEYWORD

nonn,easy,nice,mult

AUTHOR

N. J. A. Sloane

EXTENSIONS

Additional comments from Michael Somos, May 19 2000

STATUS

approved

A321069

Greatest prime factor of n^3+2.

+10
1

3, 5, 29, 11, 127, 109, 23, 257, 43, 167, 43, 173, 733, 1373, 307, 683, 983, 2917, 2287, 4001, 157, 71, 283, 223, 5209, 47, 127, 3659, 24391, 587, 9931, 113, 433, 6551, 809, 569, 307, 27437, 433, 10667, 439, 239, 1559, 223, 91127, 16223, 4153, 457, 39217, 62501

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

D. R. Heath-Brown, The largest prime factor of x^3+2, Proceedings of the London Mathematical Society, 82:3 (2000), pp. 554-596.

Christopher Hooley, On the greatest prime factor of a cubic polynomial, Journal für die reine und angewandte Mathematik, 303 (1978), pp. 21-50.

A. J. Irving, The largest prime factor of x^3+2, arXiv:1412.0024 [math.NT], 2014.

MATHEMATICA

Table[FactorInteger[n^3 + 2] [[-1, 1]], {n, 80}] (* Vincenzo Librandi, Oct 27 2018 *)

PROG

(Magma) [Maximum(PrimeDivisors(n^3 + 2)): n in [1..60]]; // Vincenzo Librandi, Oct 27 2018 *)

(PARI) a(n) = vecmax(factor(n^3+2)[, 1]); \\ Michel Marcus, Oct 27 2018

CROSSREFS

Greatest prime factors of polynomials: A006530 (n), A076565 (2n+1), A076566 (3n+3), A076567 (4n+6), A164314 (n^2-2), A076605 (n^2-1), A014442 (n^2+1), A069902 (n^2+n), A074399 (n^2+n), A199423 (2n^2+n), A089619 (2n^2+2n+1), A037464 (4n^2-1), A253254 (9n^2-7n), A093074 (n^3-n), A081257 (n^3-1), A081256 (n^3+1), A321069(n^3+2), A281793 (n^3+n^2+n+1), A281793 (n^4-1), A096172 (n^4+1), A190136 (n^4 + 6n^3 + 11n^2 + 6n), A140538 (2n^4+1), A240548 (n^5+1), A281794 (n^5+n^3+n^2+1), A240549 (n^6+1), A240550 (n^7+1), A240551 (n^8+1), A240552 (n^9+1), A240553 (n^10+1).

KEYWORD

nonn

AUTHOR

Charles R Greathouse IV, Oct 27 2018

STATUS

approved

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